# How do you find the value of the discriminant and determine the nature of the roots x^2 + 9 = 6x?

Jul 12, 2016

$\Delta = 0$

The given quadr. eqn. has two equal roots.

#### Explanation:

The discriminant $\Delta$ of a Quadratic Eqn. $: a {x}^{2} + b x + c = 0$ is given by,

$\Delta = {b}^{2} - 4 a c$.

The quadr. eqn. given to us is $: {x}^{2} + 9 = 6 x$
$\implies {x}^{2} - 6 x + 9 = 0.$

Therefore, $a = 1 , b = - 6 , c = 9$
$\Rightarrow \Delta = {\left(- 6\right)}^{2} - 4 \cdot 1 \cdot 9 = 36 - 36 = 0$

$\Rightarrow$ The given quadr. eqn. has two equal roots.

Jul 12, 2016

double-root at x = 3

#### Explanation:

y = x^2 - 6x + 9 = (x - 3)^2
There is a double-root at x = 3. The parabola is tangent to x-axis
at x = 3. Vertex (3, 0)